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PowerShell's Clear-History doesn't clear history

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<t>PowerShell's Clear-History doesn't clear history</t>

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<t>tl;dr

There are two histories to clear:
PowerShell's own, invariably in-session-only history (use Clear-History to clear it)
Additionally, in consoles (terminals), that of the PSReadLine module that is used for command-line editing by default in PowerShell v5+, which provides both an in-session and a persistent history (to clear the in-session history only, use [Microsoft.PowerShell.PSConsoleReadLine]::ClearHistory())
Alt-F7 in PSReadLine v1.2+ performs both calls for you, but note that the persistent history is unaffected (see next point).
In versions 1.2+ of PSReadLine (verify with Get-Module PSReadLine) pressing Alt+F7 performs both calls for you, and therefore fully clears the in-session history.
However, it does not clear the persistent history that has accumulated up to this point, so even the cleared session's history will resurface in future sessions by default.
To also clear the persistent (saved-to-a-file) history, you currently have to manually delete the underlying file (Remove-Item (Get-PSReadlineOption).HistorySavePath), as discussed below, and as wrapped by the Clear-SavedHistory function in the bottom section.
GitHub issue #25933 is a proposal to implement a new cmdlet that makes deleting the persistent history (too) easier. 
 
To complement CB.'s helpful answer and JVimes's helpful answer:
PowerShell's own history mechanism (Get-History, Clear-History) is host-independent, which is why - somewhat unexpectedly - you also need to clear the hosts's command history separately.
As for the console host's own history feature:
doskey-style history feature, before module PSReadline shipped with PowerShell (see below):
There is no saved history - a history is kept onl 
 
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